Here we prove that counting maximum matchings in planar, bipartite graphs is #P-complete. This is somewhat surprising in the light that the number of perfect matchings in planar graphs can be computed in polynomial time. We also prove that counting non-necessarily perfect matchings in planar graphs is already #P-complete if the problem is restricted to bipartite graphs. So far hardness was proved only for general, non-necessarily bipartite graphs.

翻译：在此, 我们证明在平面、 双部分图形中计算最大匹配值是 # P- 已完成的 。 这有点令人惊讶, 因为平面图形中的完美匹配数可以用多元时间来计算 。 我们还证明, 如果问题仅限于双部分图形, 平面图形中计算非必然完美的匹配数已经是 # P- 完成 。 因此, 远处的硬度只能用普通的、 非必然的双部分图形来证明 。